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Stats The Normal Distribution 6.4-6.6 Normally Distributed Variables • Density curves—the curve that represents the distribution – A density curve is always above or on the horizontal axis (can’t have a negative frequency) – The TOTAL area under the density curve is 1 • *The percentage of all possible observations of the variable that lie within a specific range equals (approximately) the corresponding area under the density curve, expressed as a percentage The Normal Curve (Bell Curve) • Lots of things are “approximately” normal (follow the general normal curve). • A normal curve is completely determined by the mean and standard deviation (known as parameters) • Pg 297 figure 6.21 and 6.22 Standardizing • We standardize or normalize to get an even ground for comparison…when standardized: 1 0 and • When we standardize a variable, we get the standard normal curve (not just a normal curve) • Standardize by finding z-scores for the variable (a z-score is an expression of how many standard deviations the value is away from the mean) . z x Normal Compared to Standard Normal • Standard normal curves are normal curves that have been shifted so the mean is at 0 and the standard deviation is 1 (a non standard normal curve can have any value for a mean and standard deviation). • We can use standard normal curves to find the percentages of ANY normal distribution that lie between two values. • We can use standard normal curves to compare different normal distributions Area Under the Standard Normal Curve • Standard Normal Curve Properties (z-curve) – Total area under is 1 – Extends indefinitely in both directions – Symmetric about 0 (mean) has a standard deviation of 1 – Almost all the area under the standard normal curve lies between -3 and 3 (this would be the three standard deviation rule) Table C • Standard normal curve is so important that it z-scores has its own table… Corresponding area to the LEFT of the z-score Examples • • • • • • Pg 304 ex 6.29 Pg 304 ex 6.30 Pg 305 ex 6.31 Pg 307 ex 6.34 Pg 308 ex 6.35 Pg 309 ex 6.36 – If the area we need is not exactly in table C, then we pick the closest area in the table and use that z-score. If the desired area is exactly in the middle of two areas in table C, we take the mean of the two corresponding z-scores and use that value. 68.26—95.44—99.74 Emperical Rule • Similar to the 3-standard deviation rule, but this is specific for a normal distribution • Pg 298 figure 6.24 • Procedure for finding an observed value for a specific percentage – – – – Sketch normal curve Shade region of interest Use table C to determine desired z-score(s) Find the x-values (using z-score formula) for desired zscores z z This notation is used to denote the z-score that has an area of α to the RIGHT (this notation is used in a later formula) Find z0.3: So z0.3 is wanting to know the z-score that has an area of 0.3 to the right Working With Normally Distributed Variables • General guidelines – 1. sketch the normal curve – 2. shade the area of interest and mark its x-values – 3. find the corresponding z-scores – 4. Use table C to find the desired area • Pg 314 ex6.39